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Full text of "Mathematical And Physical Papers - Iii"

68     ON  THE EFFECT OF THE  INTERNAL  FRICTION OF  FLUIDS
motion in their most general shape, which shews very clearly the effect of friction in continually consuming a portion of the work of the forces acting on the fluid.
Let P19 P9, P3 be the three normal, and T19 T2, T8 the three tangential pressures in the direction of three rectangular planes parallel to the co-ordinate planes, and let D be the symbol of differentiation with respect to t when the particle and not the point of space remains the same. Then the general equations applicable to a heterogeneous fluid (the equations (10) of my former paper) are
..(132),
with the two other equations which may be written down from symmetry.    The pressures Plt &c. are given by the equations
and four other similar equations.    In these equations oc,    du    dv    dw
V= fjjp (i** + tf + w*) dx dy dz,
At the end of the time t let V be the vis viva of a limited portion of the fluid, occupying the space which lies inside the closed surface S, and let V+DVbs the vis viva of the same mass at the end of the time t + Dt. Then
..... (135),
the triple integrals extending throughout the space bounded by S. Substituting now for jr- , &c. their values given by the equations of the system (132), we get DV =
+ vY+wZ) dx dy dz
-Wt